Jindrich Zapletal
Whenever P is a proper definable forcing for adding a real, the countable support iteration of P has all the preservation properties it can possibly have, within a wide syntactically identified class of properties.
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We prove that any countable support iteration formed with posets with $ω_2$-p.i.c.\ has $ω_2$-c.c., assuming CH in the ground model and assuming also that $ω_1$ is not collapsed. This improves earlier results of Shelah by removing the restriction on the length of the iteration. Thus, we solve the problem of obtaining a large continuum via such forcing iterations.
The preservation theorems for semi-properness, hemi-properness, and pseudo-completeness hold for countable support iterations as well as revised countable support iterations, notwithstanding the fact that the "factor lemma" fails for the countable support versions. Example: The countable support iteration of Namba forcing over a ground model of CH does not add reals. Example: A countable support iteration built from Namba forcing and Cohen forcing, over a ground model of CH, does not collapse omega_1
We show that many countable support iterations of proper forcings preserve Souslin trees. We establish sufficient conditions in terms of games and we draw connections to other preservation properties. We present a proof of preservation properties in countable support interations in the so-called Case A that does not need a division into forcings that add reals and those who do not.
We present a method for iterating semiproper forcing which uses side conditions and is inspired by the technique recently introduced by Neeman.
Shelah introduced the revised countable support (RCS) iteration to iterate semiproperness. This was an endpoint in the search for an iteration of a weak condition, still implying that aleph1 is preserved. Dieter Donder found a better manageable approach to this iteration, which is presented here.
We describe the countable ordinals in terms of iterations of Mostowski collapsings. This gives a proof-theoretic bound of definable countable ordinals in the Zermelo-Fraenkel's set theory ZF.
Limit Filters and Dependent Choice in Countable-Support Symmetric Iterations
本文研究了可数支持对称迭代中极限滤子的构造及其在ZF和DC中的应用，证明了特定模型满足ZF和DC，并展示了ω₁-完备滤子对控制AC失败程度的重要性。
We show if we use countable support iteration of forcing notions not adding reals that satisfy additional conditions, then the limit forcing does not add reals. As a result we prove that we can amalgamate two earlier methods and prove the consistency with ZFC + GCH of two statements gotten separately earlier: Souslin hypothesis and non-club guessing. We also answer a question of Justin Moore by proving the consistency of one further case of "strong failure of club guessing" with GCH.
The main result of the present paper is that $\mathfrak a_g$, the minimal size of maximal cofinitary group, can be of countable cofinality. To prove this we define a natural poset for adding a maximal cofinitary group of a given cardinality, which enjoys certain combinatorial properties allowing it to be used within a similar template forcing construction. Additionally we obtain that $\mathfrak a_p$, the minimal size of a maximal family of almost disjoint permutations, and $\mathfrak a_e$, the minimal size of a maximal eventually different family, can be of countable cofinality.
A countable-support symmetric iteration separating PP from AC
本文通过可数支持对称迭代构造一个满足ZF+DC+PP+AC_{wo}+¬AC的对称模型，利用轨道对称化包和局部分裂原理实现PP与AC的分离。
Iteration theorems for subversions of forcing classes
针对Jensen提出的子真和子完全力迫相关类，证明了修正可数支撑迭代下三类子真力迫的可迭代性，还通过宫本良迭代理论去掉子完全/子真性的技术条件，仍保可迭代性等性质并推广类似定理。
We deal with an iteration theorem of forcing notion with a kind of countable support of nice enough forcing notion which is proper aleph_2-c.c. forcing notions. We then look at some special cases (Q_D 's preceded by random forcing).
Tanaka (1997) proved a powerful generalization of Friedman's self-embedding theorem that states that given a countable nonstandard model $(\mathcal{M},\mathcal{A})$ of the subsystem $\mathrm{WKL}_{0}$ of second order arithmetic, and any element $m$ of $\mathcal{M}$, there is a self-embedding $j$ of $(\mathcal{M},\mathcal{A})$ onto a proper initial segment of itself such that $j$ fixes every predecessor of $m$. Here we extend Tanaka's work by establishing the following results for a countable nonstandard model $(\mathcal{M},\mathcal{A})$ of $\mathrm{WKL}_{0} $ and a proper cut $\mathrm{I}$ of $…
We investigate the relationship between (countable) transfinite iteration and ordinal arithmetic. The nice connection between finite iteration and addition, multiplication, and exponentiation is lost when passing to the transfinite. In this note, we investigate a new equivalence relation on ordinal functionals with respect to which we restore the connection.
Symmetric Iterations with Countable and $<κ$-Support: A Framework for Choiceless ZF Extensions
本文提出了一种基于对称迭代的框架，用于扩展选择性ZF，并在不同支持条件下实现了对称构造和DC的保持。
Comments on Choiceless Chain Conditions
本文针对无选择公理下的可数链条件，证明若其有限支持迭代仍成立则需可数可数集族选择公理，修正相关定义问题，同时研究Knaster原理与该条件的关系。
Shelah shows that certain revised countable support (RCS) iterations do not add reals. His motivation is to establish the independence (relative to large cardinals) of Avraham's problem on the existence of uncountable non-constuctible sequences all of whose proper initial segments are constructible, Friedman's problem on whether every 2-coloring of $S^2_0=\{α<ω_2\colon\cf(α)=ω\}$ has an uncountable sequentially closed homogeneous subset, and existence of a precipitous normal filter on $ω_2$ with $S^2_0\in{\cal F}$. The posets which Shelah uses in these constructions are Prikry forcing, Namba f…
In Part I of this series, we introduced a class of notions of forcing which we call Sigma-Prikry, and showed that many of the known Prikry-type notions of forcing that center around singular cardinals of countable cofinality are Sigma-Prikry. We showed that given a Sigma-Prikry poset P and a P-name for a non-reflecting stationary set T, there exists a corresponding Sigma-Prikry poset that projects to P and kills the stationarity of T. In this paper, we develop a general scheme for iterating Sigma-Prikry posets and, as an application, we blow up the power of a countable limit of Laver-indestruc…
Distinguishing Martin's axiom from its restrictions
该研究提出一种对强着色破坏极小的可数链条件力迫迭代方法，证明了Martin公理严格强于两类受限形式的Martin公理，给出了更精细的公理强度区分。
Let αbe a countable ordinal and ¶(α) the collection of its subsets isomorphic to α. We show that the separative quotient of the set ¶(α) ordered by the inclusion is isomorphic to a forcing product of iterated reduced products of Boolean algebras of the form P(ω^γ)/I(ω^γ), where γis a limit ordinal or 1 and I(ω^γ) the corresponding ordinal ideal. Moreover, the poset ¶(α) is forcing equivalent to a two-step iteration P(ω)/Fin * π, where πis an ω_1-closed separative pre-order in each extension by P(ω)/Fin and, if the distributivity number is equal toω_1, to P(ω)/Fin. Also we analyze the quotients…
We show how to force, with finite conditions, the forcing axiom PFA(T), a relativization of PFA to proper forcing notions preserving a given Souslin tree T. The proof uses a Neeman style iteration with generalized side conditions consisting of models of two types, and a preservation theorem for such iterations. The consistency of this axiom was previously known by the standard countable support iteration, using a preservation theorem due to Miyamoto.
I prove preservation theorems for countable support iteration of proper forcing concerning certain classes of capacities and submeasures. New examples of forcing notions and connections with measure theory are included.
We study the spectrum of forcing notions between the iterations of $σ$-closed followed by ccc forcings and the proper forcings. This includes the hierarchy of $α$-proper forcings for indecomposable countable ordinals as well as the Axiom A forcings. We focus on the bounded forcing axioms for the hierarchy of $α$-proper forcings and connect them to a hierarchy of weak club guessing principles. We show that they are, in a sense, dual to each other. In particular, these weak club guessing principles separate the bounded forcing axioms for distinct countable indecomposable ordinals. In the study o…
We introduce a simplified framework for ord-transitive models and Shelah's non elementary proper (nep) theory. We also introduce a new construction for the countable support nep iteration.
There is a proper countable support iteration of length $ω$ adding no new reals at finite stages and adding a Sacks real in the limit.
It is consistent that there is a partial order (P,<) of size aleph_1 such that every monotone (unary) function from P to P is first order definable in (P,<). The partial order is constructed in an extension obtained by finite support iteration of Cohen forcing. The main points is that (1) all monotone functions from P to P will (essentially) have countable range (this uses a Delta-system argument) and (2) that all countable subsets of P will be first order definable, so we have to code these countable sets into the partial order. Amalgamation of finite structures plays an essential role.
Using countable support iteration of $S$-proper posets, for some appropriate stationary set $S$, we obtain a generic extension of the constructible universe, in which $\mathfrak{b}=\mathfrak{c}=\aleph_2$ and there is a maximal cofinitary group with a $Π^1_2$-definable set of generators.
Iterating Generalised Perfect Set Forcing Along Well-Founded Orders
提出几何力迫迭代技术的推广版本，证明沿任意良基偏序可迭代广义完美集力迫并保持基数至κ⁺，应用于树状力迫保持ℵ₁。
We show that splitting forcing does not have the weak Sacks property below any condition, answering a question of Laguzzi, Mildenberger and Stuber-Rousselle. We also show how some partition results for splitting trees hold or fail and we determine the value of cardinal invariants after an $ω_2$-length countable support iteration of splitting forcing.
We give a self-contained proof of the preservation theorem for proper countable support iterations known as "tools-preservation," "Case A" or "first preservation theorem" in the literature. We do not assume that the forcings add reals.
We provide a general preservation theorem for preserving selective independent families along countable support iterations. The theorem gives a general framework for a number of results in the literature concerning models in which the independence number $\mathfrak{i}$ is strictly below $\mathfrak{c}$, including iterations of Sacks forcing, Miller partition forcing, $h$-perfect tree forcings, coding with perfect trees. Moreover, applying the theorem, we show that $\mathfrak{i} = \aleph_1$ in the Miller Lite model. An important aspect of the preservation theorem is the notion of "Cohen preserva…
We show that in a model obtained by forcing with a countable support iteration of length omega_2 of Mathias forcing h(2), the distributivity number of r.o.(P(omega)/fin)^2, is omega_1 but h, the one of P(omega)/fin, is omega_2.
We show that in the aleph_2-stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(omega)/fin. This complements Vojtas' result, that under cf(c)=p the two algebras are isomorphic
We present reasons for developing a theory of forcing notions which satisfy the properness demand for countable models which are not necessarily elementary submodels of some (H(chi), in). This leads to forcing notions which are ``reasonably'' definable. We present two specific properties materializing this intuition: nep (non-elementary properness) and snep (Souslin non-elementary properness). For this we consider candidates (countable models to which the definition applies), and the older Souslin proper. A major theme here is ``preservation by iteration'', but we also show a dichotomy: if suc…
We introduce a class of notions of forcing which we call $Σ$-Prikry, and show that many of the known Prikry-type notions of forcing that centers around singular cardinals of countable cofinality are $Σ$-Prikry. We show that given a $Σ$-Prikry poset $\mathbb P$ and a name for a non-reflecting stationary set $T$, there exists a corresponding $Σ$-Prikry poset that projects to $\mathbb P$ and kills the stationarity of $T$. Then, in a sequel to this paper, we develop an iteration scheme for $Σ$-Prikry posets. Putting the two works together, we obtain a proof of the following. Theorem. If $κ$ is the…
Given a Polish space X and a countable family of analytic hypergraphs on X, I consider the sigma-ideal generated by Borel sets which are anticliques in at least one hypergraph in the family. It turns out that many of the quotient posets are proper. I investigate the forcing properties of these posets, certain natural operations on them, and prove some related dichotomies. For this broad class of posets, most fusion arguments and iteration preservation arguments can be replaced with simple combinatorial considerations concerning the hypergraphs.
A general theory of iterated forcing using finitely additive measures
本文基于有限可加测度发展了一般迭代力迫理论，引入 $μ$-$\mathrm{FAM}$-linked 和广义交数等新概念，证明其保持强无界族与反 Bendixson 族，并用于证明 Cichoń 图左侧的新分离结果。
Property B: A Baumgartner-style Property that Applies to Preservation of $\aleph_1$ and $\aleph_2$ under Iterations with Supports of Size $\aleph_1$
本文研究了在支持大小为$\aleph_1$的迭代中保持$\aleph_1$和$\aleph_2$的性质B，扩展到其他满足$κ^{<κ}=κ$的基数，并引入了修正的完美集forcing。
Forcing over choiceless models and generic absoluteness
该研究开发了无选择集合论模型上的力迫工具，规避了可数链条件力迫坍缩ω₁的问题，还证明了特定力迫类下的通用绝对性原则蕴含无限基数均有可数共尾性。
We show that after forcing with a countable support iteration or a finite product of Sacks or splitting forcing over $L$, every analytic hypergraph on a Polish space admits a $\mathbfΔ^1_2$ maximal independent set. As a main application we get the consistency of $\mathfrak{r} = \mathfrak{u} = \mathfrak{i} = ω_2$ together with the existence of a $Δ^1_2$ ultrafilter, a $Π^1_1$ maximal independent family and a $Δ^1_2$ Hamel basis. This solves open problems of Brendle, Fischer and Khomskii and the author. We also show in ZFC that $\mathfrak{d} \leq \mathfrak{i}_{cl}$.
We introduce a class of proper posets which is preserved under countable support iterations, includes $ω^ω$-bounding, Cohen, Miller, and Mathias posets associated to filters with the Hurewicz covering properties, and has the property that the ground model reals remain splitting and unbounded in corresponding extensions. Our results may be considered as a possible path towards solving variations of the famous Roitman problem.
A virtual five element basis for the uncountable linear orders
该研究通过证明Aronzsajn线与Countryman线的同构嵌入结论，结合真力迫等新保持性结果，解答了不可数线性序五元基存在性等相关数理逻辑问题，推广了前人工作。
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